It used a trinomial tree model. As prices change from one period to another they can move in three different ways. Up, down or stable.

The problem is, rates were negative which caused the probability of moving up and down to sum up to more than one.

Which meant that the probability of staying stable was... that's right... negative. Probabilities need to add up to one, but should never be negative!

The model didn't describe how the world actually is, but showed us how it 'ought' to be. And then crashed. Over and over again.

Dipping A Toe Into Absurdity

When events defy old boundaries of reasonableness, the model maker updates what's possible. Before we lived in a world where negative interest rates were a fantasy, now we're living in a world where that's possible and we fix the model to accomodate these new realities.

There is another possibility however.

Espen Haug explores how negative probabilities can let models themselves open the door to such absurd worlds.

Common garden variety probabilities should be between one and zero, and the sum of outcomes sum up to one.

Summing up events with negative probabilities also sum up to one. Allowing for negative probabilities also results in outcomes which have a probability of more than one.

Negative probabilities give us an expanded canvas as new unforeseen events occur, without having to manually open up the model, take out a monkey wrench to expand the sample space.

You may retort, but what does it mean to have a minus ten percent chance of seeing negative interest rates?

Don't know for sure.

Perhaps it means our original canvas of event possibilities was 100% and now we have found 20% more canvas. Our 'model world' has grown by a fifth from our initial starting point.

The Half-Coin

A rather more prosaic example for those of you not familiar or interested with financial modelling is based on a flip of a coin.

Take a coin and toss it twice, add the result.

We have a one in four chance of seeing two zeroes.

A fifty fifty chance of seeing a one. I.e. we can 'expect'

And another one in four chance of seeing two ones.

We can described flipping two coins as:

0.25*0+0.5*1+0.25*2 = 1

This is called a probability generating function (PGF) and it adds up to 1.

The PGF of a single coin flip is:

0.5*0 + 0.5*1 = 1

which coincidentally is the square root of the previous two coin case.